Kähler structures for holomorphic submersions

In this short paper, for any holomorphic submersion $\pi :X\to B$, we derive a criterion for $X$ to have Kähler structures. This criterion generalizes Blanchard’s criterion for a special class of isotrivial holomorphic submersions. We use this criterion to answer a question of Harvey-Lawson in the case of fiber dimension one. As the main application, we prove that the existence of Hermitian-Symplectic structures on certain class of holomorphic submersions with Kähler fibers and Kähler bases implies that the total spaces are Kähler. This class includes isotrivial submersions and torus fibrations.